Subgroups — #28376235

Order 13952.

1540 conjugacy classes of subgroups.

The subgroup lattice has 142 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295509	C₂	1	8	yes	maximal, complemented
2	#20295504	C₂	1	2	yes	maximal, complemented
3	#20295501	C₂	1	6	yes	maximal, complemented
4	#41×Dic₁₀₉	C₂	1	8	yes	maximal, complemented
5	C₂×#16812658	C₂	1	6	yes	maximal, complemented
6	C₂²×#41×C₁₀₉	C₂	1	1	yes	maximal, characteristic
7	Q₈×Dic₁₀₉	C₂²	1	16	yes	complemented
8	D₄×Dic₁₀₉	C₂²	1	48	yes	complemented
9	#16812658	C₂²	1	4	yes	—
10	#16812658	C₂²	1	12	yes	—
11	#16812658	C₂²	1	12	yes	complemented
12	#16812658	C₂²	1	12	yes	complemented
13	C₂×C₄⋊Dic₁₀₉	C₂²	1	12	yes	complemented
14	C₂×C₂²⋊Dic₁₀₉	C₂²	1	12	yes	—
15	C₂×C₄×Dic₁₀₉	C₂²	1	12	yes	complemented
16	C₂×#41×C₁₀₉	C₄	1	16	yes	complemented
17	C₂×#41×C₁₀₉	C₂²	1	4	yes	—
18	C₂×#41×C₁₀₉	C₂²	1	4	yes	—
19	C₂²×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
20	C₂²×D₄×C₁₀₉	C₂²	1	3	yes	—
21	C₂³×C₄×C₁₀₉	C₂²	1	3	yes	—
22	C₂²×Dic₁₀₉	C₂³	1	24	yes	—
23	C₄⋊Dic₁₀₉	C₂³	1	8	yes	—
24	C₄⋊Dic₁₀₉	C₂³	1	24	yes	—
25	C₂²⋊Dic₁₀₉	C₂³	1	8	yes	—
26	C₂²⋊Dic₁₀₉	C₂³	1	24	yes	—
27	C₄×Dic₁₀₉	C₂³	1	8	yes	—
28	C₄×Dic₁₀₉	C₂³	1	24	yes	—
29	C₂⁴×C₁₀₉	C₂³	1	3	yes	—
30	#41×C₁₀₉	C₂×C₄	1	32	yes	complemented
31	#41×C₁₀₉	C₂×C₄	1	32	yes	complemented
32	C₂×Q₈×C₁₀₉	C₂×C₄	1	8	yes	complemented
33	C₂×Q₈×C₁₀₉	C₂³	1	4	yes	—
34	C₂×D₄×C₁₀₉	C₂×C₄	1	24	yes	complemented
35	C₂×D₄×C₁₀₉	C₂³	1	12	yes	—
36	C₂²×C₄×C₁₀₉	C₂×C₄	1	24	yes	complemented
37	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
38	C₂²×C₄×C₁₀₉	C₂³	1	3	yes	—
39	C₂²×C₄×C₁₀₉	C₂³	1	6	yes	—
40	C₂²×C₄×C₁₀₉	C₂³	1	6	yes	—
41	C₂³×C₁₀₉	—	4	6	no	—
42	C₂³×C₁₀₉	C₂²×C₄	1	12	yes	—
43	C₂³×C₁₀₉	C₂⁴	1	1	yes	characteristic
44	C₂³×C₁₀₉	C₂⁴	1	6	yes	—
45	Q₈×C₁₀₉	C₂²×C₄	1	16	yes	—
46	D₄×C₁₀₉	C₂²×C₄	1	48	yes	—
47	C₂×C₄×C₁₀₉	C₂²×C₄	1	4	yes	—
48	C₂×C₄×C₁₀₉	C₂²×C₄	1	12	yes	—
49	C₂×C₄×C₁₀₉	C₂²×C₄	1	24	yes	—
50	C₂×C₄×C₁₀₉	C₂²×C₄	1	24	yes	—
51	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
52	C₂×C₄×C₁₀₉	C₂⁴	1	6	yes	—
53	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
54	C₂×Dic₁₀₉	—	4	24	no	—
55	C₂×Dic₁₀₉	C₂⁴	1	16	yes	—
56	C₂²×C₁₀₉	—	2	12	no	—
57	C₂²×C₁₀₉	—	2	2	no	—
58	C₂²×C₁₀₉	—	4	12	no	—
59	C₂²×C₁₀₉	C₂³×C₄	1	2	yes	—
60	C₂²×C₁₀₉	C₂³×C₄	1	12	yes	—
61	C₂²×C₁₀₉	C₂⁵	1	1	yes	characteristic
62	C₄×C₁₀₉	C₂³×C₄	1	2	yes	—
63	C₄×C₁₀₉	C₂³×C₄	1	2	yes	—
64	C₄×C₁₀₉	C₂³×C₄	1	12	yes	—
65	Dic₁₀₉	—	2	16	no	—
66	C₂×C₁₀₉	—	2	2	no	—
67	C₂×C₁₀₉	—	2	12	no	—
68	C₂×C₁₀₉	C₂⁴×C₄	1	1	yes	characteristic
69	C₂×C₁₀₉	#585	1	2	yes	—
70	#3428	—	109	1	no	maximal, Sylow
71	C₁₀₉	#3428	1	1	yes	characteristic, complemented, Sylow
72	C₂²×#41	D₁₀₉	1	1	yes	characteristic
73	#520	—	109	8	no	—
74	#519	—	109	2	no	—
75	#518	—	109	6	no	—
76	C₄×#41	—	109	8	no	—
77	C₂×#117	—	109	6	no	—
78	C₂×#41	Dic₁₀₉	1	16	yes	complemented
79	C₂×#41	C₂×D₁₀₉	1	4	yes	—
80	C₂×#41	C₂×D₁₀₉	1	4	yes	—
81	C₂²×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
82	C₂²×D₄	C₂×D₁₀₉	1	3	yes	—
83	C₂³×C₄	C₂×D₁₀₉	1	3	yes	—
84	C₄×Q₈	—	109	16	no	—
85	C₄×D₄	—	109	48	no	—
86	#117	—	109	4	no	—
87	#117	—	109	12	no	—
88	#117	—	109	12	no	—
89	#117	—	109	12	no	—
90	C₂×C₄⋊C₄	—	109	12	no	—
91	C₂×C₂²⋊C₄	—	109	12	no	—
92	C₂×C₄²	—	109	12	no	—
93	C₂⁴	C₂²×D₁₀₉	1	3	yes	—
94	#41	C₂×Dic₁₀₉	1	32	yes	complemented
95	#41	C₂×Dic₁₀₉	1	32	yes	complemented
96	C₂×Q₈	C₂×Dic₁₀₉	1	8	yes	complemented
97	C₂×Q₈	C₂²×D₁₀₉	1	4	yes	—
98	C₂×D₄	C₂×Dic₁₀₉	1	24	yes	complemented
99	C₂×D₄	C₂²×D₁₀₉	1	12	yes	—
100	C₂²×C₄	—	109	24	no	—
101	C₂²×C₄	C₂×Dic₁₀₉	1	24	yes	complemented
102	C₂²×C₄	C₂²×D₁₀₉	1	6	yes	—
103	C₂²×C₄	C₂²×D₁₀₉	1	3	yes	—
104	C₂²×C₄	C₂²×D₁₀₉	1	6	yes	—
105	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
106	C₄⋊C₄	—	109	8	no	—
107	C₄⋊C₄	—	109	24	no	—
108	C₂²⋊C₄	—	109	8	no	—
109	C₂²⋊C₄	—	109	24	no	—
110	C₄²	—	109	24	no	—
111	C₄²	—	109	8	no	—
112	C₂³	—	4	6	no	—
113	C₂³	C₂²×Dic₁₀₉	1	12	yes	—
114	C₂³	C₂³×D₁₀₉	1	1	yes	characteristic
115	C₂³	C₂³×D₁₀₉	1	6	yes	—
116	Q₈	C₂²×Dic₁₀₉	1	16	yes	—
117	D₄	C₂²×Dic₁₀₉	1	48	yes	—
118	C₂×C₄	—	109	16	no	—
119	C₂×C₄	—	436	24	no	—
120	C₂×C₄	C₂²×Dic₁₀₉	1	4	yes	—
121	C₂×C₄	C₂²×Dic₁₀₉	1	12	yes	—
122	C₂×C₄	C₂²×Dic₁₀₉	1	24	yes	—
123	C₂×C₄	C₂²×Dic₁₀₉	1	24	yes	—
124	C₂×C₄	C₂³×D₁₀₉	1	6	yes	—
125	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
126	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
127	C₂²	—	2	12	no	—
128	C₂²	—	2	2	no	—
129	C₂²	—	4	12	no	—
130	C₂²	C₂³×Dic₁₀₉	1	2	yes	—
131	C₂²	C₂³×Dic₁₀₉	1	12	yes	—
132	C₂²	C₂⁴×D₁₀₉	1	1	yes	characteristic
133	C₄	—	218	16	no	—
134	C₄	C₂³×Dic₁₀₉	1	2	yes	—
135	C₄	C₂³×Dic₁₀₉	1	2	yes	—
136	C₄	C₂³×Dic₁₀₉	1	12	yes	—
137	C₂	—	2	2	no	—
138	C₂	—	2	12	no	—
139	C₂	C₂⁴×Dic₁₀₉	1	1	yes	characteristic
140	C₂	#20295862	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂³×D₁₀₉
commutator	C₂×C₁₀₉	C₂⁴×C₄
Frattini	C₂²	C₂⁴×D₁₀₉
Fitting, prime radical	C₂²×#41×C₁₀₉	C₂
socle, cosocle	C₂²×C₁₀₉	C₂⁵
hypercenter	C₂²×#41	D₁₀₉
hypocenter	C₁₀₉	#3428
LCS	G ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×#41

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. #20295509	C₂	semidirect
2. #20295504	C₂	semidirect
3. #20295501	C₂	semidirect
4. #41×Dic₁₀₉	C₂	semidirect
5. C₂×#16812658	C₂	semidirect
7. Q₈×Dic₁₀₉	C₂²	semidirect
8. D₄×Dic₁₀₉	C₂²	semidirect
11. #16812658	C₂²	semidirect
12. #16812658	C₂²	semidirect
13. C₂×C₄⋊Dic₁₀₉	C₂²	semidirect
15. C₂×C₄×Dic₁₀₉	C₂²	semidirect
16. C₂×#41×C₁₀₉	C₄	semidirect
30. #41×C₁₀₉	C₂×C₄	semidirect
31. #41×C₁₀₉	C₂×C₄	semidirect
32. C₂×Q₈×C₁₀₉	C₂×C₄	semidirect
34. C₂×D₄×C₁₀₉	C₂×C₄	semidirect
36. C₂²×C₄×C₁₀₉	C₂×C₄	semidirect
71. C₁₀₉	#3428	semidirect
78. C₂×#41	Dic₁₀₉	semidirect
94. #41	C₂×Dic₁₀₉	semidirect
95. #41	C₂×Dic₁₀₉	semidirect
96. C₂×Q₈	C₂×Dic₁₀₉	semidirect
98. C₂×D₄	C₂×Dic₁₀₉	semidirect
101. C₂²×C₄	C₂×Dic₁₀₉	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#20295509	2
#20295504	2
#20295501	2
#41×Dic₁₀₉	2
C₂×#16812658	2
C₂²×#41×C₁₀₉	2
#3428	109